Phonon-mediated spin-flipping mechanism in the spin ices Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$

TL;DR

This study investigates phonon-mediated spin-flipping mechanisms in spin ices Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$, providing a quantitative model for high-temperature spin dynamics supported by experimental evidence.

Contribution

It introduces a constrained model of phonon-mediated spin flipping that explains high-temperature spin dynamics in these materials, supported by experimental coupling evidence.

Findings

Quantitative model of phonon-mediated spin flipping

Experimental evidence of crystal field and phonon coupling

Model accurately describes high-temperature spin processes

Abstract

To understand emergent magnetic monopole dynamics in the spin ices Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$, it is necessary to investigate the mechanisms by which spins flip in these materials. Presently there are thought to be two processes: quantum tunneling at low and intermediate temperatures and thermally activated at high temperatures. We identify possible couplings between crystal field and optical phonon excitations and construct a strictly constrained model of phonon-mediated spin flipping that quantitatively describes the high-temperature processes in both compounds, as measured by quasielastic neutron scattering. We support the model with direct experimental evidence of the coupling between crystal field states and optical phonons in Ho$_2$Ti$_2$O$_7$.

Equations

$$\Gamma (T)=\sum _i\frac{3\pi r{n}_i}{2M{\Delta }_i}{\zeta }_{\mu }^2{Z}_{\mu }({\Delta }_i)\{|⟨a|Q_{\mu }|v_i⟩{|}^2+|⟨b|Q_{\mu }|v_i⟩{|}^2\},$$

$$\frac{{\zeta }_i}{{\zeta }_3}=\sqrt{\frac{B_i{\Delta }_i{Z}_{\text{O}(48f)}({\Delta }_3)\left[{\sum }_{\mu }{\sum }_{j=\pm }|⟨j|Q(\mu )|v_3^j⟩{|}^2\right]}{B_3{\Delta }_3{Z}_{\text{O}(48f)}({\Delta }_i)\left[{\sum }_{\mu }{\sum }_{j=\pm }|⟨j|Q(\mu )|v_i^j⟩{|}^2\right]}},$$

$$\frac{Z_{\text{O}(48f)}(\Delta =21\text{meV})}{Z_{\text{O}(48f)}(\Delta =60\text{meV})}=\frac{Z_{\text{O}(48f)}(\Delta =26\text{meV})}{Z_{\text{O}(48f)}(\Delta =60\text{meV})}=1.5$$

$$\frac{Z_{\text{O}(48f)}(\Delta =31\text{meV})}{Z_{\text{O}(48f)}(\Delta =60\text{meV})}=1.$$

Full text

spin ice

In rare earth compounds, magnetic responses can be strongly and non-monotonically dependent on the strength or frequency of applied magnetic field, or the temperature. Examples include stepped magnetization curves in single ion magnets Ishikawa et al. (2005), or the multiply-peaked susceptibility response in LiYF4:Ho3+ Giraud et al. (2001, 2003); Bertaina et al. (2006). These effects appear because there are competing mechanisms that can contribute to the flipping of large rare earth magnetic moments. Owing to their different origins - conduction electrons Becker et al. (1977), phonon-mediated (e.g. direct, Raman, Orbach, and phonon bottleneck effects Orbach (1961); Finn et al. (2002); Scott and Jeffries (1962)), or quantum mechanical (tunneling, thermally assisted tunneling between excited states, resonant tunneling at electronic-nuclear level crossings, and co-tunneling Thomas et al. (1996); Bokacheva et al. (2000); Giraud et al. (2001, 2003); Gatteschi and Sessoli (2003)) - these mechanisms have quite different parametric dependencies. Understanding spin flipping (or relaxation) is currently important in rare-earth based single ion magnets Blagg et al. (2013), especially in the context of applications in quantum information processing Leuenberger and Loss (2001); Ardavan et al. (2007); Bogani and Wernsdorfer (2008) that depend on the stability and control of quantum states Prokof’ev and Stamp (2000); Feynmann and Vernon Jr (1963); Ghosh et al. (2002); Stamp and Tupitsyn (2004), and in spin ices, where they determine the mobility of magnetic monopole excitations Castelnovo et al. (2008).

In a canonical spin ice such as Dy2Ti2O7 or Ho2Ti2O7 Bramwell and Gingras (2001), the magnetization dynamics of the low temperature Coulomb phase Fennell et al. (2009); Henley (2010) should be described by the cooperative behavior of the thermal population of emergent magnetic monopoles Castelnovo et al. (2008); Ryzhkin (2005), which form a magnetic Coulomb gas. Indeed, the spin relaxation time, $\tau$, of Dy2Ti2O7, as extracted from $\chi_{ac}$, has been explained with considerable success by the monopole picture: both a thermally activated regime at $T<1$ K Snyder et al. (2001); Matsuhira et al. (2001); Snyder et al. (2003, 2004) (which we call low temperature) and temperature independent plateau for $1<T<10$ K (intermediate temperature) are captured well by a theory of monopole hopping in dilute (unscreened) and concentrated (screened) magnetic Coulomb gases respectively Jaubert and Holdsworth (2009). The reentrant low temperature thermal activation is due to interactions between unscreened monopoles.

For a monopole to hop, a spin must be flipped, and because the plateau of $\tau$ was previously associated with quantum tunneling of the large, Ising-like Dy3+ moments between the members of their ground state doublet Snyder et al. (2004), monopoles were assumed to hop by tunneling of the spins with temperature independent attempt frequency Jaubert and Holdsworth (2009). The resulting picture should describe the Coulomb gas realized in each material by relating the energy for monopole creation and unbinding to the exchange interactions Brooks-Bartlett et al. (2014); Zhou et al. (2011). However, subsequent measurements of Ho2Ti2O7 Quilliam et al. (2011) and Dy2Ti2O7 Matsuhira et al. (2011); Yaraskavitch et al. (2012); Kassner et al. (2015) have found that in the unscreened regime this relationship is not exactly as expected, while simulations of Dy2Ti2O7 with a temperature-dependent hop rate agree better with the observed relaxation times Takatsu et al. (2013). These studies suggest that to understand out-of-equilibrium Giblin et al. (2011); Revell et al. (2012); Paulsen et al. (2014, 2016) and quantum dynamics Tomasello et al. (2015); Rau and Gingras (2015) in spin ices at low temperature, it is essential to understand all contributions to the monopole hopping dynamics. As in LiYF4:Ho3+ Bertaina et al. (2006), the first requirement is to understand the classical spin flipping mechanism of the spin ices.

Studies of Dy2Ti2O7 Snyder et al. (2001); Matsuhira et al. (2001); Snyder et al. (2003, 2004) in which the intermediate temperature plateau was ascribed to quantum tunneling of the spins also revealed a second thermally activated regime for $T>10$ K (i.e. high temperature). The response of Ho2Ti2O7 is similar but the relative rate of the low temperature process is much faster than in Dy2Ti2O7 Ehlers et al. (2002); sup . The high temperature process in both spin ices was modeled by an Arrenhius law, with activation energy $\Delta$ attributed to over-barrier hopping via the first crystal field excitation (CFE). However, the best-fitting $\Delta$, although close, is not equal to the energy of any CFE in either material, and this interpretation does not explain how such a process would occur.

We propose that phonon mediated processes involving a higher crystal field state interacting with phonons Lovesey and Staub (2000) provide a quantitative and physical explanation of the high temperature processes. In this mechanism, a rare earth ion is excited from one crystal field state to an intermediate excited state by absorption of a phonon, and then relaxes to a third state by emission of another phonon. Relaxation by a single such process has the characteristic temperature dependence of $n=1/(\exp{(\Delta/k_{B}T)}-1)$, where $\Delta$ is the energy of the phonon to be absorbed, but more than one process can operate simultaneously, depending which crystal field levels interact with phonons. The time and temperature scales of this type of process mean they can be studied by neutron scattering. Either the width ($\Gamma$) of the quasielastic neutron scattering (QENS) can be understood as lifetime broadening of the ground state doublet and used to give a measure of the spin relaxation time ($\tau$), as was done for rare earth cuprates Lovesey and Staub (2000); or the width of a CFE can be followed directly, as was done for LiTmF4 Babkevich et al. (2015). In the former case the origin of the relaxation was debated Boothroyd (2001), while in the latter full details of the coupled phonons were not established. In the following, we measure $\Gamma$ using QENS, determine the allowed spin-lattice interactions and construct a model of phonon mediated processes in both materials that describes the high temperature processes quantitatively. We provide microscopic evidence of one such coupling.

We have measured QENS in powder samples of both Ho2Ti2O7 and Dy2Ti2O7 sup over a wide range of temperatures using the spectrometer FOCUS Jurányi et al. (2003) at SINQ. We report results obtained using the $(0,0,2)$ reflection of both the pyrolytic graphite ($\lambda=4.3,5,6$ Å; resolution $\approx 100,50,40$ $\mu$eV) and mica ($\lambda=11$ Å; resolution $\approx 20$ $\mu$eV) monochromators, where we selected the wavelength to give appropriate resolution for a range of temperatures. The quasielastic scattering was fitted by a single Lorentzian, adjusted by the Bose factor for detailed balance. The elastic line was removed by fitting with a Gaussian peak, whose parameters were derived from a measurement of the resolution using a vanadium standard. Additional Lorentzians were incorporated in the fit of high temperature data from Ho2Ti2O7 to model excited state CFEs that appear close to the elastic line. Points at the edges of two resolution ranges were measured with both settings to ensure overlap of the fitted peak widths.

An example of the $S(|\vec{Q}|,\omega)$ data obtained for Ho2Ti2O7 is shown in Fig. 1a, and an example of the $|\vec{Q}|$-integrated data used for fitting is shown in Fig. 1b. The temperature dependence of the width and intensity of the quasielastic scattering and CFEs can be seen in Fig. 1c. Below $T\approx 50$ K, the spin fluctuation processes are too slow for QENS, and, even with $\lambda=11$ Å, the response is resolution-limited, but as the temperature is further increased, the QENS broadens.

In Fig. 2a, we show the temperature evolution of $\Gamma$ for both compounds, and also its representation as a relaxation time $\tau$ (panel b). Notably, the QENS spectra of Dy2Ti2O7 are nearly twice as broad as those of Ho2Ti2O7 throughout the sampled temperature range. In Fig. 2c,d we show our data for Ho2Ti2O7 compared with NSE data from Ref. Ehlers et al. (2002), which extends to longer times/lower temperatures, in terms of $\Gamma$ and $\tau$ respectively. All the lines in Fig. 2 are derived from models, either the model which we discuss below, or the Arrenhius law used in Ref. Ehlers et al. (2004). It can be seen in Ref. Ehlers et al. (2004) that the relaxation time already departs from the Arrhenius law at the highest temperatures studied there, and this is made plain by the higher temperatures measured in this work (see Fig. 2c and d).

The phonon-mediated spin relaxation mechanism depends on a magnetoelastic interaction of normal modes of vibration with the single ion crystal field potential Lovesey and Staub (2000). The contribution to the temperature dependence of $\Gamma$ is given by

[TABLE]

where $\zeta_{\mu}$ is the magnetoelastic coupling parameter for a phonon and intermediate crystal field state $|v_{i}\rangle$ at energy $\Delta_{i}$, $r$ is the number of ions per unit cell, and $M$ the mass of an oxide ion. The distribution function $n_{i}=(\exp{(\Delta_{i}/k_{B}T)}-1)^{-1}$ provides the temperature dependence of the process, and $Z_{\mu}(\Delta_{i})$ is the partial phonon density of states (pPDOS) of the anionic modes of vibration transforming according to the representation $\mu$. The prefactors are absorbed into fitting parameters such that $\Gamma(T)=\sum_{j}B_{i}n_{i}$ Lovesey and Staub (2000).

For an intermediate state $i$ to enter the summation, we require a finite matrix element for the quadrupolar operator ($Q_{\mu}$) for the transition between the initial ($\langle a|$) and intermediate ($|v\rangle$) crystal field state and spectral overlap of this state with a phonon ($u_{\mu}$) of identical symmetry ($\mu$ labels the irreducible representation of the operator or excitation). At the rare earth site, in $D_{3d}$ symmetry, there are three quadrupolar operators with symmetry $(A_{1},E,E)$, and the matrix element for a transition $\langle a|Q_{\mu}|v\rangle$ is finite if the direct product of the representations ($\gamma$) of the two states and the transition operator contain the unit representation, $\gamma_{a}\times\gamma_{v}\times\gamma_{Q}\in A_{1}$. There are three possible combinations for finite matrix elements of the magnetoelastic interaction operator: $\gamma_{a,Q,v,u}=E$ (1); $\gamma_{a,v}=E,\gamma_{Q,u}=A$ (2); $\gamma_{a,Q,u}=E,\gamma_{v}=A$ (3). The initial state $|a\rangle$ is a member of the ground state doublet, and $|v\rangle$ is an excited crystal field state, the final state $|b\rangle$ is the other member of the ground state doublet.

The transitions involved in the model are summarized in Fig. 3. Using the wave functions of crystal field states Ruminy et al. (2016a), we evaluated the quadrupolar matrix elements of the crystal field transitions. Quasi-degenerate (at the Brillouin zone center) phonon modes of the correct symmetry were identified from the phonon band structure Ruminy et al. (2016b). The vibrational modes involved are dominated by oxide ions in the $48f$ position, so we approximate $Z_{\mu}(\Delta_{i})$ by the pPDOS of this site.

In Ho2Ti2O7, we find the two largest matrix elements between the ground state and the doublets at $E=26.3,60$ meV, and weaker matrix elements between the ground state and the singlet and doublet at $E=21,22$ meV. Each of these transitions is quasi-degenerate with a phonon of appropriate symmetry, while the remaining transitions have no overlap with a vibrational mode of $E$ or $A$ symmetry. For Ho2Ti2O7, we construct our model using three intermediate states at $\Delta_{i}=21.5,26.3,60$ meV, where the first represents the effect of the weak matrix elements. In Dy2Ti2O7, the intermediate states are those at $E=21,31,43$ meV. The state at 91 meV also meets the symmetry requirements but is outside the temperature window of this study. Other states have large matrix elements, but no compatible phonon.

For Ho2Ti2O7 we included both QENS and NSE data in the fit, and since the states $B_{1}$ and $B_{2}$ have similar energies and nearly identical pPDOS Ruminy et al. (2016b), we related their values by the ratio of their quadrupolar matrix elements. The resulting coefficients are $B_{1,2,3}=0.018,0.2,0.79$ meV. For Dy2Ti2O7, to reduce the number of fitting parameters, the values of the parameters $B_{2}$ and $B_{3}$ were linearly related using the energies of their CFEs, unity for the ratio of the oxygen phonon density of states, and their quadrupolar transition matrix elements. We obtained $B_{1,2,3}=0.23,0.49,0.38$ meV. As shown in Fig. 2, the model fits the relaxation rates of both compounds very well. For Ho2Ti2O7, relaxation via the level at 26.3 meV describes the QENS width effectively up to $T\approx 250$ K, and the third intermediate state at $E=60$ meV dominates at higher temperatures.

The values for the magnetoelastic coupling constants $\zeta_{\mu}$ extracted from the fitted parameters under these approximations sup suggest that the magnetoelastic coupling is linear (in energy), consistent with physical ingredients of the model. To further verify our model, we sought direct evidence of interactions between CFEs and phonons using a single crystal of Ho2Ti2O7 and the triple axis spectrometer HB3 at HFIR, ORNL. The $(0,0,2)$ reflection of the beryllium monochromator provides access to quite high energy transfers with good energy resolution - using a pyrolytic graphite filter and analyzer ($(0,0,2)$ reflection), the energy resolution was $\Delta E\approx 1.7$ meV in the energy transfer window of $20-30$ meV. With fixed final energies of $E_{f}=14.7,30.5$ meV, we measured energy scans in the range $18<E<33$ meV at different $(0,0,l)$ positions that were either Brillouin zone centers $(0,0,l=2,4,6,8,10,12)$ or boundaries $(0,0,l=3,5,7,9)$, at $T=5,200$ K.

Fig. 4a shows the two CFEs at $E\approx 22,26.3$ meV and a phonon at $E\approx 31$ meV, measured at $(0,0,8)$. The intensities of the CFEs decrease as the temperature is raised, and they shift upward in energy, while the intensity of the phonon increases but its energy does not change. The upward shift of the CFEs is also shown by the downward shift of the excited state transitions T1 and T2 in Fig. 1c. Comparison of the same scan at $l=7,8$ shows a resolution limited sharp peak for both CFEs with a 0.5 meV upward dispersion between zone boundary and center for the first, but at identical energies for the second. The $l$-dependence of the intensity of the CFEs (Fig. 4b) follows the dipole magnetic form factor at zone boundaries ($l=n$) and at zone centers where the $Fd\bar{3}m$ space group forbids a Bragg reflection ($l=2n$), but the CFE at $E\approx 22$ meV has anomalously large intensity at zone centers with strong Bragg reflections ($l=4n$) while the CFE at $E\approx 26.3$ meV also follows the magnetic form factor at these positions.

Phonon calculations Ruminy et al. (2016b) show that there is an optical phonon with $E$ symmetry at $E\approx 22$ meV at the zone center. The phonon disperses away at the zone boundary, and its structure factor is suppressed at zone centers where the Bragg intensity is not allowed. Hence at all these positions (i.e. $l=n$ and $l=2n$) both CFEs are unaffected and follow the magnetic form factor. At those zone centers with a strong Bragg reflection, the strong phonon structure factor boosts the intensity well above the magnetic form factor, but the observation of a single mode displaced from the energy of the uncoupled zone boundary CFE or phonon shows that the coupling pulls the two excitations into resonance, i.e. they are not just coincident. Conversely, the phonon mode expected to interact with the CFE at 26.3 meV was calculated to have a very weak structure factor along $(0,0,l)$, due to its polarization. Hence we observe no signatures of coupling in this direction, and this CFE also follows the magnetic form factor (Fig. 4c).

We have shown that the symmetries and wavefunctions CFEs and optical phonons can be used to construct a physically realistic model for phonon mediated spin flipping processes. Modes with the correct symmetry and energy exist in the spin ices Ho2Ti2O7 and Dy2Ti2O7, and we presented direct evidence of one of the couplings in Ho2Ti2O7. A model based on these spin-lattice interactions describes the high temperature spin relaxation in both compounds very well. We advance this model as the first microscopic description of a spin flipping mechanism in the spin ices Ho2Ti2O7 and Dy2Ti2O7, and also as a quantification of the spin-lattice interactions possible in these materials. Our investigation sets the stage for microscopic investigations of the possible quantum processes at low temperature, and their consequences for collective monopole dynamics.

I Supplementary Information

Further details of samples and experiments: The powder samples of Ho2Ti2O7 and Dy2Ti2O7 were originally described in Refs. Ruminy et al. (2016b, a). They were prepared from stochiometric ratios of the oxides Ho2O3 or Dy2O3, and TiO2 in a solid state reaction. The oxides, with 99.99% purity, were annealed at 850 oC for 10 hours, then mixed and ground, and heated at 950-1300 oC for 140 hours with several intermediate grindings. The structures were verified by combined neutron and x-ray diffraction experiments, which were carried out on HRPT Fischer et al. (2000) at SINQ, PSI and the Materials Science Beamline (MSB) Willmott et al. (2013) at the SLS, PSI. Rietveld refinement of the structures as implemented in the Fullprof Rodriguez-Carvajal (1993) software proved both samples to be of high quality and single phase.

For the QENS experiment on FOCUS, to overcome the strong absorption of natural isotopic abundance dysprosium we used a flat plate sample holder with very thin layer of sample. The plate was oriented perpendicular to the incident beam to reduce the path length of neutrons in the sample, and we used data from low angle detectors only. To have reasonable statistics requires long counting times. We also confirmed some points by measuring our isotopically enriched crystal of 162Dy2Ti2O7, but the crystal is much smaller than the cross section of the beam so intersects relatively few incident neutrons and hence does not greatly improve the situation. Ho2Ti2O7 can be used directly as a packed powder and the count rate is much larger.

Magnetoelastic coupling constants: The determination of an absolute value for the magnetoelastic coupling constant $\zeta$ from the fitted $B$ parameters is difficult. Despite having the full calculated pyrochlore phonon spectrum Ruminy et al. (2016b), there is a large uncertainty connected with the evaluation of the partial phonon densities of states $Z_{\mu}(\Delta_{i})$, which only includes individual vibrational modes that transform according to the representation $\mu$. We note however, that all vibrational modes involved in the model in both compounds are dominated by the motion of the oxygen O(48$f$) ions, which surround the rare earth ion. Hence, the partial phonon density of states of the O(48$f$) ions gives at least good estimates for the ratios of the $Z_{\mu}(\Delta_{i})$ at different energy transfers $\Delta_{i}$. We therefore estimate all the magnetoelastic coupling constants $\zeta_{i}$ for the intermediate CEF states in both Ho2Ti2O7 and Dy2Ti2O7 relative to the coupling constant $\zeta_{3}$ of the third intermediate CEF state in Ho2Ti2O7 at $E=60$ meV:

[TABLE]

where the summation $j$ runs over the two members of the corresponding ground state doublet and all members of the excited intermediate CEF state $v_{i}$. The ratios of the partial phonon densities of states are taken as:

[TABLE]

[TABLE]

In order to compare between two different ions, we calculate the ratios of the bare magnetoelastic coupling constants $\tilde{\zeta_{i}}$, which follow from $\zeta=\tilde{\zeta}\alpha{J}$ Lovesey and Staub (2000) with the Stevens factors $\alpha(J)$ of the rare earth ions, here $\alpha=-1/450$ and $-2/315$ for Ho3+ and Dy3+ respectively. The resulting ratios are illustrated in Fig. 5. We find that the bare magnetoelastic coupling constants of all intermediate CEF states in the two compounds depend linearly on the energy transfer of the intermediate states within errorbars. Given a linear mode coupling such as between the quadrupolar operators and the normal modes of vibration in the magnetoelastic interaction operator of Eq. 1, it is plausible that the magnetoelastic coupling strength vanishes in the limit of zero energy transfer and is generally highest at high energy transfer. The experimental observations are therefore consistent with the expectation and give strong support for the phonon mediated mechanism with the established intermediate CEF states in both rare earth pyrochlore titanates Ho2Ti2O7 and Dy2Ti2O7.

Relation with previously described relaxation processes: Previously two relaxation processes have been reported in both compounds Snyder et al. (2001); Matsuhira et al. (2001); Snyder et al. (2003, 2004); Ehlers et al. (2002). In Dy2Ti2O7 these can be seen in $\chi_{ac}$ experiments because the relative rates of them are favorable for this technique. The low temperature process, ascribed to quantum tunneling due to the plateau in the relaxation time at intermediate temperatures, is sufficiently slow that it does not compete with the high temperature process even as this becomes slow at temperatures of $T\approx 20$ K. Hence peaks can be observed in $\chi^{\prime\prime}$ (as measured by $\chi_{ac}$) as each process dominates the relaxation in the experimental frequency range, first the high temperature process at $T\approx 20$ K, then the low temperature process at $T<10$ K. In Ho2Ti2O7, the low temperature process seems to be relatively much faster, and competes with the high temperature process effectively on the time scales of $\chi_{ac}$. Therefore, one cannot observe a peak in $\chi^{\prime\prime}$ that corresponds to the high temperature process using $\chi_{ac}$ measurements. To observe the high temperature process one must go to higher temperatures where the low temperature process is no longer competitive and the high temperature process is necessarily much faster. This means that techniques with shorter timescales are required to observe it, which is why it was originally found using neutron spin echo and quasielastic neutron scattering Ehlers et al. (2002, 2004).

The high temprature process in Dy2Ti2O7 was fitted by an Arrhenius law with activation energy $\Delta=17-25$ meV, which was compared to the first CFE. At this time, the exact energy of the CFE was not actually known, but was expected to be of this order. It is now known to be at $21$ meV Ruminy et al. (2016a). In Ho2Ti2O7, the activation energy was given as $\Delta=25(\pm 1)$ meV, but the CFEs are at $E=21.9,26.3$ meV. The high temperature process was described as a flipping of the spin via the first CFE, but in this case the activation energy should match exactly the energy of the CFE, and the Arrhenius law should describe the relaxation time throughout the temperature range where the process is dominant. Neither of these conditions is actually fulfilled, particularly when the relaxation time is measured to higher temperatures.

However, the previous investigations showed that the presence or absence of a peak in $\chi^{\prime\prime}_{ac}$ depends on the relative rate of high and low temperature processes. In absence of a competing low temperature process Ho2Ti2O7 would also have a peak in $\chi^{\prime\prime}_{ac}$ for the high temperature process at a similar temperature to Dy2Ti2O7. This condition is still satisfied if the temperature dependence of the relaxation rate provided by our model is treated in an analogous fashion. Hence the phonon-mediated model provides a physical explanation and quantitative description of all known observations about the high temperature processes.